TY - JOUR
T1 - Influential coalitions for boolean functions I
T2 - Constructions
AU - Bourgain, Jean
AU - Kahn, Jeff
AU - Kalai, Gil
N1 - Publisher Copyright:
© 2024 Jean Bourgain, Jeff Kahn, and Gil Kalai.
PY - 2024
Y1 - 2024
N2 - For f : {0, 1}n → {0, 1} and S ⊂ {1, 2, . . .,,n}, let J+ S(f) be the probability that, for x uniform from {0. 1}n, there is some y ϵ {0, 1}n with f(y) = 1 and x = y outside S. We are interested in estimating, for given μ(f) (:= E(f)) and m, the least possible value of max {J+ s(f): |S| = m}. A theorem of Kahn, Kalai, and Linial (KKL) gave some understanding of this issue and led to several stronger conjectures. Here we disprove a pair of conjectures from the late 80s, as follows.
AB - For f : {0, 1}n → {0, 1} and S ⊂ {1, 2, . . .,,n}, let J+ S(f) be the probability that, for x uniform from {0. 1}n, there is some y ϵ {0, 1}n with f(y) = 1 and x = y outside S. We are interested in estimating, for given μ(f) (:= E(f)) and m, the least possible value of max {J+ s(f): |S| = m}. A theorem of Kahn, Kalai, and Linial (KKL) gave some understanding of this issue and led to several stronger conjectures. Here we disprove a pair of conjectures from the late 80s, as follows.
KW - Boolean functions
KW - Discrete isoperimetric inequalities
KW - Influence
KW - Trace of set-systems
KW - Tribes
UR - http://www.scopus.com/inward/record.url?scp=105008972582&partnerID=8YFLogxK
U2 - 10.4086/toc.2024.v020a004
DO - 10.4086/toc.2024.v020a004
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AN - SCOPUS:105008972582
SN - 1557-2862
VL - 20
JO - Theory of Computing
JF - Theory of Computing
ER -